We propose a phase fleld model that approximates its limiting sharp interface model (free boundary problem) up to second order in interface thickness. A broad range of double-well potentials can be utilized so long as the dynamical coe-cient in the phase equation is adjusted appropriately. This model thereby assures that computation with particular value of interface thickness , will difier at most by O(2) from the limiting sharp interface problem. As an illustration, the speed of a traveling wave of the phase fleld model is asymptotically expanded to demonstrate that it difiers from the speed of the traveling wave of the limit problem by O( 2 ). 1. Introduction. Interface problems arising from solidiflcation have been studied extensively in mathematics and physics for more than a century (29). The math- ematical study began with Lamµe and Clapeyron (23) in 1831 who modeled the freezing of the ground using the heat equation, latent heat across the interface, and the condition that the temperature at the interface remains at the equilibrium freez- ing temperature. Reformulated in 1889, this problem became known as the classical Stefan model (31), and can be stated as follows. Determine the temperature, T(x;t); and the interface, i(t); satisfying the system of equations, ‰cTt = K ¢T in ›ni(t); ‰'vn = K ((rT ¢ N)) i ; T(x;t) = TE on i(t); with '; c; as the latent heat and heat capacity per unit mass, K the difiusivity, ‰ the density and TE the equilibrium freezing temperature. The unit normal and velocity at the interface are given by N and vn, and ((¢¢¢)) + denotes the difierence in the limits from the two sides of the interface. The classical Stefan model has the appealing mathematical feature that the temperature, T(x;t); determines the phase at each point (x;t): By deflnition, T(x;t) > TE implies the material is liquid at that point (or, more generally, in the phase with the higher internal energy), while T(x;t) < TE means it is solid, while T(x;t) = TE deflnes the interface i(t): Thus, the condition that T(x;t) = TE at the interface appears to be mathematically convenient. The mathematical study of the classical Stefan model posed di-cult challenges that